# Wave6 Third Layer: Stabilizers, Bolzano-Weierstrass, Compact Subsets, Total Expectation, Binomial Theorem, Mobius Function and CFGs Complete seven traceable knowledge objects in dependency order on the existing number-system and function foundations. This is a suggested learning sequence, not a universal logical dependency or a formal proof. - Stage 1 · required ## [Finite group action](/en/mko/mko-finite-group-action/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261109%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-finite-group-action) [Concept guides](/en/derivations/?q=mko-finite-group-action) - Stage 1 · required ## [Subgroups and group homomorphisms](/en/mko/mko-subgroup-homomorphism/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261109%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-subgroup-homomorphism) [Concept guides](/en/derivations/?q=mko-subgroup-homomorphism) - Stage 1 · required ## [Subsequence](/en/mko/mko-subsequence/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261109%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-subsequence) [Concept guides](/en/derivations/?q=mko-subsequence) - Stage 1 · required ## [Sequence limits and exact tail cutoffs](/en/mko/mko-sequence-limit/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261109%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-sequence-limit) [Concept guides](/en/derivations/?q=mko-sequence-limit) - Stage 1 · required ## [Real number](/en/mko/mko-real-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261109%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-real-number) [Concept guides](/en/derivations/?q=mko-real-number) - Stage 1 · required ## [Supremum and infimum](/en/mko/mko-supremum-and-infimum/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261109%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-supremum-and-infimum) [Concept guides](/en/derivations/?q=mko-supremum-and-infimum) - Stage 1 · required ## [Compact topological space](/en/mko/mko-compact-topological-space/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261109%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-compact-topological-space) [Concept guides](/en/derivations/?q=mko-compact-topological-space) - Stage 1 · required ## [Subspace topology](/en/mko/mko-subspace-topology/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261109%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-subspace-topology) [Concept guides](/en/derivations/?q=mko-subspace-topology) - Stage 1 · required ## [Finite conditional expectation](/en/mko/mko-conditional-expectation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261109%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-conditional-expectation) [Concept guides](/en/derivations/?q=mko-conditional-expectation) - Stage 1 · required ## [Expectation in finite models](/en/mko/mko-expectation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261109%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-expectation) [Concept guides](/en/derivations/?q=mko-expectation) - Stage 1 · required ## [Event](/en/mko/mko-event/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261109%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-event) [Concept guides](/en/derivations/?q=mko-event) - Stage 1 · required ## [Permutations and combinations](/en/mko/mko-permutation-combination/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261109%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-permutation-combination) [Concept guides](/en/derivations/?q=mko-permutation-combination) - Stage 1 · required ## [Mathematical induction](/en/mko/mko-mathematical-induction/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261109%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-mathematical-induction) [Concept guides](/en/derivations/?q=mko-mathematical-induction) - Stage 1 · required ## [Univariate polynomials over the rationals](/en/mko/mko-polynomial/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261109%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-polynomial) [Concept guides](/en/derivations/?q=mko-polynomial) - Stage 1 · required ## [Prime natural number](/en/mko/mko-prime-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261109%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-prime-number) [Concept guides](/en/derivations/?q=mko-prime-number) - Stage 1 · required ## [Unique prime factorization of integers](/en/mko/mko-unique-prime-factorization/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261109%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-unique-prime-factorization) [Concept guides](/en/derivations/?q=mko-unique-prime-factorization) - Stage 1 · required ## [Function mapping](/en/mko/mko-function-mapping/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261109%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-function-mapping) [Concept guides](/en/derivations/?q=mko-function-mapping) - Stage 1 · required ## [Abstract computational model](/en/mko/mko-computational-model/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261109%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-computational-model) [Concept guides](/en/derivations/?q=mko-computational-model) - Stage 1 · required ## [Set](/en/mko/mko-set/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261109%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-set) [Concept guides](/en/derivations/?q=mko-set) - Stage 1 · required ## [Sequence](/en/mko/mko-sequence/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261109%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-sequence) [Concept guides](/en/derivations/?q=mko-sequence) - Stage 2 · optional ## [Stabilizer of a finite group action](/en/mko/mko-finite-group-stabilizer/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261109%2F) The stabilizer of a specified point consists of every group element fixing that point and forms a subgroup. [Related questions](/en/explore/?mko=mko-finite-group-stabilizer) [Concept guides](/en/derivations/?q=mko-finite-group-stabilizer) - Stage 3 · optional ## [Bolzano-Weierstrass theorem for real sequences](/en/mko/mko-bolzano-weierstrass-theorem-real/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261109%2F) Every whole bounded real sequence has a convergent subsequence selected by an infinite strictly increasing index map. [Related questions](/en/explore/?mko=mko-bolzano-weierstrass-theorem-real) [Concept guides](/en/derivations/?q=mko-bolzano-weierstrass-theorem-real) - Stage 4 · optional ## [Compact subset](/en/mko/mko-compact-subset/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261109%2F) A subset K of X is compact in its subspace topology exactly when every relative-open cover has a finite subcover. [Related questions](/en/explore/?mko=mko-compact-subset) [Concept guides](/en/derivations/?q=mko-compact-subset) - Stage 5 · optional ## [Law of total expectation](/en/mko/mko-law-of-total-expectation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261109%2F) On the same finite probability space, weighting every partition-cell conditional mean by its cell probability gives E[X]. [Related questions](/en/explore/?mko=mko-law-of-total-expectation) [Concept guides](/en/derivations/?q=mko-law-of-total-expectation) - Stage 6 · optional ## [Binomial theorem](/en/mko/mko-binomial-theorem/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261109%2F) With commutative multiplication, a binomial power of natural exponent n has a complete finite expansion with binomial coefficients. [Related questions](/en/explore/?mko=mko-binomial-theorem) [Concept guides](/en/derivations/?q=mko-binomial-theorem) - Stage 7 · optional ## [Mobius function](/en/mko/mko-mobius-function/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261109%2F) On positive integers, complete prime multiplicities determine the Mobius function: mu(1)=1, a repeated prime gives0, and a squarefree integer has the parity sign. [Related questions](/en/explore/?mko=mko-mobius-function) [Concept guides](/en/derivations/?q=mko-mobius-function) - Stage 8 · optional ## [Context-free grammar](/en/mko/mko-context-free-grammar/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261109%2F) A CFG uses finite disjoint nonterminal and terminal alphabets, a start symbol and finite productions with a single nonterminal on each left side. [Related questions](/en/explore/?mko=mko-context-free-grammar) [Concept guides](/en/derivations/?q=mko-context-free-grammar) Path source & statuspath-pulse-foundations-20261109 · candidate [Original paths JSON](/data/architecture/learning-paths.json)